The Mystery of Tau – Miskolczi – Part Five – Equation Soufflé

In previous articles in this series we looked at a number of issues first in Miskolczi’s 2010 paper and then in the 2007 paper.

The author himself has shown up and commented on some of these points, although not all, and sadly decided that we are not too bright and a little bit too critical and better pastures await him elsewhere.

The 2007 paper by Ferenc Miskolczi is a soufflé of confusion piled on confusion. Sorry to be blunt. If I was writing a paper I would say “..some clarity is needed in important sections..” but many readers unfamiliar with the actual meaning of this phrase might think that some clarity was needed in important sections rather than the real truth that the paper is a shambles.

I’ll refer to this paper as M2007. And to the equations in M2007 with an M prefix – so, for example, equation 15 will be [M15].

Some background is needed so first we need to take a look at something called The Semi-Gray Model. Regular readers will find a lot of this to be familiar ground, but it is necessary as there are always many new readers.

The SGM – Semi-Grey Model or Schwarzschild Grey Model

I’ll introduce this by quoting from an excellent paper referenced by M2007. This is a 1995 paper by Weaver and the great Ramanathan (free link in References):

Simple models of complex systems have great heuristic value, in that their results illustrate fundamental principles without being obscured by details. In particular, there exists a long history of simple climate models. Of these, radiative and radiative-convective equilibrium models have received great attention..

One of the simplest radiative equilibrium models involves the assumption of a so-called grey atmosphere, where the absorption coefficient is assumed to be independent of wavelength. This was first discussed by Schwarzschild [1906] in the context of stellar interiors. The grey gas model was adapted to studies of the Earth by assuming the atmosphere to be transparent to solar radiation and grey for thermal radiation. We will refer to this latter class as semigrey models.

And in the abstract they say:

Radiative equilibrium solutions are the starting point in our attempt to understand how the atmospheric composition governs the surface and atmospheric temperatures, and the greenhouse effect. The Schwarzschild analytical grey gas model (SGM) was the workhorse of such attempts. However, the solution suffered from serious deficiencies when applied to Earth’s atmosphere and were abandoned about 3 decades ago in favor of more sophisticated computer models..

[Emphasis added]

And they go on to present a slightly improved SGM as a useful illustrative tool.

Some clarity on a bit of terminology for new readers – a blackbody is a perfect emitter and absorber of radiation. In practice there are no blackbodies but some bodies come very close. A blackbody has an emissivity = 1 and absorptivity = 1.

In our atmosphere, the gases which absorb and emit significant radiation have very wavelength dependent properties, e.g.:

From spectralcalc.com

Figure 1

So the emissivity and absorptivity vary hugely from one wavelength to the next (note 1). However, as an educational tool, we can calculate the results for a grey atmosphere – this means that the emissivity is assumed to be constant across all wavelengths.

The term semi-grey means that the atmosphere is considered transparent for shortwave = solar wavelengths (<4 μm) and constant but not zero for longwave = terrestrial wavelengths (>4 μm).

We assume that the atmosphere is in radiative equilibrium – that is, convection does not exist and so only radiation moves heat around.

Here is a graphic showing the main elements of the model:

Figure 2

Once each layer in the atmosphere is in equilibrium, there is no heating or cooling – this is the definition of equilibrium. This means energy into the layer = energy out of the layer. So we can use simple calculus to write some equations of radiative transfer.

We define the TOA (top of atmosphere) to be where optical thickness, τ=0, and it increases through the atmosphere to reach a maximum of τ=τA at the surface. This is conventional.

We also know two boundary conditions, because at TOA (top of atmosphere) the downward longwave flux, F↓(τ=0) = 0 and the upwards longwave flux, F↑(τ=0) = F0, where F0 = absorbed solar radiation ≈ 240 W/m². This is because energy leaving the planet must be balanced by energy being absorbed by the planet from the sun.

We also have to consider the fact that energy is not just going directly up and down but is going up and down at every angle. We can deal with this via the dffusivity approximation which sums up the contributions from every angle and tells us that if we use τ*= τ . 5/3 (where τ is defined in the vertical direction) we get the complete contribution from all of the different directions. (Note 2). For those following M2007 I have used τ* to be his τ with a ˜ on top, and τ to be his τ with a ¯ on top.

With these conditions we can get a solution for the SGM (see derivation in the comments):

B(τ) = F0/2π . (τ+1) [1] cf eqn [M15]

where B is the spectrally integrated Planck function, and remember F0 is a constant.

And also:

F↑(τ) = F0/2 . (τ+2) [2]

F↓(τ) = F0/2 . τ [3]

A quick graphic might explain this a little more (with an arbitrary total optical thickness, τA* = 3):

Figure 3

Notice that the upward longwave flux at TOA is 240 W/m² – this balances the absorbed solar radiation. And the downward longwave flux at TOA is zero, because there is no atmosphere above from which to radiate. This graph also demonstrates that the difference between F↑ and F↓ is a constant as we move through the atmosphere, meaning that the heating rate is zero. The increase in downward flux, F↓, is matched by the decrease in upward flux, F↑.

It’s a very simple model.

By contrast, here are the heating/cooling rates from a comprehensive (= “standard”) radiative-convective model, plotted against height instead of optical thickness.

Heating from solar radiation, because the atmosphere is not completely transparent to solar radiation:

From Grant Petty (2006)

Figure 4

Cooling rates due to various “greenhouse” gases:

From Petty (2006)

Figure 5

And the heating and cooling rates won’t match up because convection moves heat from the surface up into the atmosphere.

Note that if we plotted the heating rate vs altitude for the SGM it would be a vertical line on 0.0°C/day.

Let’s take a look at the atmospheric temperature profile implied by the semi-grey model:

Figure 6

Now a lot of readers are probably wondering what the τ really means, or more specifically, what the graph looks like as a function of height in the atmosphere. In this model it very much depends on the concentration of the absorbing gas and its absorption coefficient. Remember it is a fictitious (or “idealized”) atmosphere. But if we assume that the gas is well-mixed (like CO2 for example, but totally unlike water vapor), and the fact that pressure increases with depth then we can produce a graph vs height:

Figure 7

Important note – the values chosen here are not intended to represent our climate system.

Figure 6 & 7, along with figure 3, are just to help readers “see” what a semi-grey model looks like. If we increase the total optical depth of the atmosphere the atmospheric temperature at the surface increases.

Note as well that once the temperature reduction vs height is too large a value, the atmosphere will become unstable to convection. E.g. for a typical adiabatic lapse rate of 6.5 K/km, if the radiative equilibrium implies a lapse rate > 6.5 K/km then convection will move heat to reduce the lapse rate.

Curious Comments on the SGM

Some comments from M2007:

p 11:

Note, that in obtaining B0 , the fact of the semi-infinite integration domain over the optical depth in the formal solution is widely used. For finite or optically thin atmosphere Eq. (15) is not valid. In other words, this equation does not contain the necessary boundary condition parameters for the finite atmosphere problem.

The B0 he is referring to is the constant in [M15]. This constant is H/2π – where H = F0 (absorbed solar radiation) in my earlier notation. This constant B0 later takes on magical properties.

p 12:

Eq. (15) assumes that at the lower boundary the total flux optical depth is infinite. Therefore, in cases, where a significant amount of surface transmitted radiative flux is present in the OLR , Eqs. (16) and (17) are inherently incorrect. In stellar atmospheres, where, within a relatively short distance from the surface of a star the optical depth grows tremendously, this could be a reasonable assumption, and Eq. (15) has great practical value in astrophysical applications. The semi-infinite solution is useful, because there is no need to specify any explicit lower boundary temperature or radiative flux parameter (Eddington, 1916).

[Emphasis added]

The equations can easily be derived without any requirement for the total optical depth being infinite. There is no semi-infinite assumption in the derivation. Whether or not some early derivations included it, I cannot say. But you can find the SGM derivation in many introductions to atmospheric physics and no assumption of infinite optical thickness exists.

When considering the clear-sky greenhouse effect in the Earth’s atmosphere or in optically thin planetary atmospheres, Eq. (16) is physically meaningless, since we know that the OLR is dependent on the surface temperature, which conflicts with the semi-infinite assumption that τA =∞..

..There were several attempts to resolve the above deficiencies by developing simple semi-empirical spectral models, see for example Weaver and Ramanathan (1995), but the fundamental theoretical problem was never resolved..

This is the reason why scientists have problems with a mysterious surface temperature discontinuity and unphysical solutions, as in Lorenz and McKay (2003). To accommodate the finite flux optical depth of the atmosphere and the existence of the transmitted radiative flux from the surface, the proper equations must be derived.

The deficiencies noted include the result in the semi-gray model of a surface air temperature less than the ground temperature. If you read Weaver and Ramanathan (1995) you can see that this isn’t an attempt to solve some “fundamental problem“, but simply an attempt to make a simple model slightly more useful without getting too complex.

The mysterious surface temperature discontinuity exists because the model is not “full bottle”. The model does not include any convection. This discontinuity is not a mystery and is not crying out for a solution. The solution exists. It is called the radiative-convective model and has been around for over 40 years.

Miskolczi makes some further comments on this, which I encourage people to read in the actual paper.

We now move into Appendix B to develop the equations further. The results from the appendix are the equations M20 and M21 on page 14.

The surface radiation attenuated by the transmittance of the atmosphere, plus..

The sum of all the contributions of atmospheric radiation – each contribution attenuated by the transmittance from that location to the top of atmosphere

For those wanting to understand the maths a little bit, the 3/2 factor that appears everywhere in Miskolczi’s equation B1 is the diffusivity approximation already mentioned (and see note 2) where we need to sum the radiances over all directions to get flux.

Now this equation is the complete equation of radiative transfer. If we combine it with a simple convective model it is very effective at calculating the flux and spectral intensity through the atmosphere – see Theory and Experiment – Atmospheric Radiation.

So equation B1 in M2007 cannot be solved analytically. This means we have to solve it numerically. This is “simple” in concept but computationally expensive because in the HITRAN database there are 2.7 million individual absorption lines, each one with a different absorption coefficient and a different line width.

However, it can be solved and that is what everyone does. Once you have the database of absorption coefficients and the temperature profile of the atmosphere you can calculate the solution. And band models exist to speed up the process.

And now the rabbit..

The author now takes the equation for the “source function” (B) from the simple model and inserts it into the “complete” solution.

The “source function” in the complete solution can already be calculated – that’s the whole point of the equation B1. But now instead, the source function from the simple model is inserted in its place. This equation assumes that the atmosphere has no convection, has no variation in emissivity with wavelength, has no solar absorption in the atmosphere, and where the heating rate at each level in the atmosphere = zero.

The origin of equation B3 is the equation you see above it:

B(τ) = 3H(τ)/4π + B0 [M13]

Actually, if you check equation M13 on p.11 it is:

B(τ) = 3H.τ/4π + B0 [M13]

This appears to be one of the sources of confusion for Miskolczi, for later comment.

Equation M13 is derived for zero heating rates throughout the atmosphere, and therefore constant H. With this simple assumption – and only for this simple assumption – the equation M13 is a valid solution to “the source function”, ie the atmospheric temperature and radiance.

If you have the complete solution you get one result. If you have the simple model you get a different result. If you take the result from one and stick it in the other how can you expect to get an accurate outcome?

If you want to see how various atmospheric factors are affected by changing τ, then just change τ in the general equation and see what happens. You have to do this via numerical analysis but it can easily be done..

As we continue on working through the appendix, B6 has a sign error in the 2nd term on the right hand side, which is fixed by B7.

This B0 is the constant in the semi-gray solution. The constant appears because we had a differential equation that we integrated. And the value of the constant was obtained via the boundary condition: upward flux from the climate system must balance solar radiation.

So we know what B0 is.. and we know it is a constant..

Yet now the author differentiates the constant with respect to τ. If you differentiate a constant it is always zero. Yet the explanation is something that sounds like it might be thermodynamics, but isn’t:

If someone wants to explain what thermodynamic principle create the first statement – I would be delighted. Without any explanation it is a jumble of words that doesn’t represent any thermodynamic principle.

Anyway B0 is a constant and is equal to approximately 240 W/m². Therefore, if we differentiate it, yes the value dB0/dτ=0.

Unfortunately, the result in B10 is wrong.

If we differentiate a variable we can’t assume it is a constant. The variable in question is BG. This is the “source function” for the ground, which gives us the radiance and surface temperature. Clearly the surface temperature is a function of many factors especially including optical thickness. Of course, if somewhere else we have proven that BG is a constant then dBG/dτ=0.

It has to be proven.

[And thanks to DeWitt Payne for originally highlighting this issue with BG, as well as explaining my calculus mistakes in an email].

A quick digression on basic calculus for the many readers who don’t like maths – just so you can see what I am getting at.. (you are the ones who should read it)

Digression

We will consider just the last term in equation [B9]. This term = BG/(eτ-1). I have dropped the π from the term to make it simpler to read what is happening.

Generally, if you differentiate two terms multiplied together, this is what happens:

d(fg)/dx = g.df/dx + f.dg/dx [4]

This assumes that f and g are both functions of x. If, for example, f is not a function of x, then df/dx=0 (this just means that f does not change as x changes). And so the result reduces to d(fg)/dx = f.dg/dx.

So, using [4] :

d/dτ [BG/(eτ-1)] = [1/(eτ-1)] . dBG/dτ + BG . d [1/(eτ-1)]/dτ [5]

We can look up:

d [1/(eτ-1)]/dτ = -eτ/(eτ-1)² [6]

So substituting [6] into [5], last term in [B9]:

= [1/(eτ-1)] . dBG/dτ – eτ.BG /(eτ-1)² [7]

You can see the 2nd half of this expression as the first term in [B10], multiplied by π of course.

But the term for how the surface radiance changes with optical thickness of the atmosphere has vanished.

end of digression

Soufflé Continued

So the equation should read:

Where the red text is my correction (see eqn 7 in the digression).

Perhaps the idea is that if we assume that surface temperature doesn’t change with optical thickness then we can prove that surface temperature doesn’t change with optical thickness.

This (flawed) equation is now used to prove B11:

Well, we can see that B11 isn’t true. In fact, even apart from the missing term in B10, the equation has been derived by combining two equations which were derived under different conditions.

As we head back into the body of the paper from the appendix, equations B7 and B8 are rewritten as equations [M20] and [M21].

Miskolczi adds:

We could not find any references to the above equations in the meteorological literature or in basic astrophysical monographs, however, the importance of this equation is obvious and its application in modeling the greenhouse effect in planetary atmospheres may have far reaching consequences.

Readers who have made it this far might realize why he is the first with this derivation.

Continuing on, more statements are made which reveal some of the author’s confusion with one part of his derivation. The SGM model is derived by integrating a simple differential equation, which produces a constant. The boundary conditions tell us the constant.

Equation [M13] is written:

B(τ) = 3H/4π + B0 [M13]

Then [M14] is written:

H(τ) = π (I+ – I-) [M14]

So now H is a function of optical depth in the atmosphere?

In [M15]:

B(τ*) = H (1 + τ*)/2π [M15]

Refer to my equation 1 and you will see they are the same. The only way this equation can be derived is with H as a constant, because the atmosphere is in radiative equilibrium. If H isn’t constant you have a different equation – M13 and 15 are no longer valid.

..The fact that the new B0 (skin temperature) changes with the surface temperature and total optical depth, can seriously alter the convective flux estimates of previous radiative-convective model computations. Mathematical details on obtaining equations 20 and 21 are summarized in appendix B.

Miskolczi has confused himself (and his readers).

Conclusion

There is an equation of radiative transfer and it is equation B1 in the appendix of M2007. This equation is successfully used to calculate flux and spectral intensity in the atmosphere.

There is a very simple equation of radiative transfer which is used to illustrate the subject at a basic level and it is called the semi-grey model (or the Schwarzschild grey model). With the last few decades of ever increasing computing power the simple models have less and less practical use, although they still have educational value.

Miskolczi has inserted the simple result into the general model, which means, at best, it can only be applied to a “grey” atmosphere in radiative equilibrium, and at worst he has just created an equation soufflé.

The constant in the simple model has become a variable. Without any proof, or re-derivation of the simple model.

One of the important variables in the simple model has become a constant and therefore vanished from an equation where it should still reside.

Many flawed thermodynamic concepts are presented in the paper, some of which we have already seen in earlier articles.

M2007 tells us that Ed=Aa due to Kirchhoff’s law. (See Part Two). His 2010 paper revised this claim as to due to Prevost.. However, the author himself recently stated:

I think I was the first who showed the Aa~=Ed relationship with reasonable quantitative accuracy.

And doesn’t understand why I think it is important to differentiate between fundamental thermodynamic identities and approximate experimental results in the theory section of a paper. “My experiments back up my experiments..”

M2007 introduces equation [M7] with:

In Eq. (6) SU − (F0 + P0 ) and ED − EU represent two flux terms of equal magnitude, propagating into opposite directions, while using the same F0 and P0 as energy sources. The first term heats the atmosphere and the second term maintains the surface energy balance. The principle of conservation of energy dictates that:
SU − (F0) + ED − EU = F0 = OLR

Note the pseudo-thermodynamic explanation. The author himself recently said:

Eq. 7 simply states, that the sum of the Su-OLR and Ed-Eu terms – in ideal greenhause case – must be equal to Fo. I assume that the complex dynamics of the system may support this assumption, and will explain the Su=3OLR/2 (global average) observed relationship.

[Emphasis added]

And later entertainingly commented:

You are right, I should have told that, and in my new article I shall pay more attantion to the full explanations. However, some scientists figured it out without any problem.

Party people who got the joke right off the bat..

M07 also introduces the idea that kinetic energy can be equated with the flux from the atmosphere to space. See Part Three. Introductory books on statistical thermodynamics tell us that flux is proportional to the 4th power of temperature, while kinetic energy is linearly proportional to temperature. We have no comment from the author on this basic contradiction.

This pattern indicates an obvious problem.

In summary – this paper does not contain a theory. Just because someone writes lots of equations down in attempt to back up some experimental work, it is not theory.

If the author has some experimental work and no theory, that is what he should present – look what I have found, I have a few ideas but can someone help develop a theory to explain these results.

Obviously the author believes he does have a theory. But it’s just equation soufflé.

Other Articles in the Series:

The Mystery of Tau – Miskolczi – introduction to some of the issues around the calculation of optical thickness of the atmosphere, by Miskolczi, from his 2010 paper in E&E

Part Two – Kirchhoff – why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paper

Part Three – Kinetic Energy – why kinetic energy cannot be equated with flux (radiation in W/m²), and how equation 7 is invented out of thin air (with interesting author comment)

Part Four – a minor digression into another error that seems to have crept into the Aa=Ed relationship

Part Six – Minor GHG’s – a less important aspect, but demonstrating the change in optical thickness due to the neglected gases N2O, CH4, CFC11 and CFC12.

Further reading:

New Theory Proves AGW Wrong! – a guide to the steady stream of new “disproofs” of the “greenhouse” effect or of AGW. And why you can usually only be a fan of – at most – one of these theories.

You assume that the global average tau (changing arbitrarily by the GHG’s) determine Bg.
This assumption contradicts to the empirical fact described in Eq. (2) which fixes Su = 3OLR/2 and maximizes g at 1/3.
According to these equations, it is OLR and the IR structure together that determine Su, and controls the amount and distribution of GHG’s to set tau.

Third,
That tau equals to the actual and the time-series average.

Fourth,
Thanks for noticing the sign error and other typos; they were corrected three years ago.

Let’s move away from Miskolczi’s theory and discuss a random theory to find out what you think about the role of theory in general.

Let’s say I seem to find that my lawn grows more when the moon is full. And less when the moon is new. It’s an exciting discovery.

I produce those results with some nice graphs.

Now I develop a theory that “explains” these results with an equation, G = k.M1.6, where G is the rate of growing, in mm/day, k is a constant and M is the phase of the moon as a fraction.

I backup my theory by developing the equation like this:

I take the standard equation for grass growing from the theory as shown in, Journal of Biology January 1951, and substitute one of the parameters in the standard theory to another arbitrary parameter that I found in a different equation, developed for different conditions, in Journal of Grass Growing, March 1972.

As a result, I produce a new equation, never before seen, then differentiate it with respect to the time of day, holding the total energy constant (without comment) to produce an equation that backs up my experimental results.

My question to you – if I can produce a theory by this mechanism to apparently support my experimental results, have I provided support for my experimental results – or not?

And if I further explain – upon being questioned by interested grass growers – that the new equations were really derived from my experimental results, have I provided theoretical support for my experimental results – or not?

Your answers will guide me in my next question.

As a preliminary heads up, it will go something like:

You claim that the theory independently supports the experiment but you invented the theory based on the experimental results and claimed it as theory. This is NOT an independent line of support and actually seems like —–!

On the specific issue of B0 and differentiating Eq 7, I understood that while B0 is constant over the range of τ for a given atmosphere, FM is differentiating with respect to τ_A – the total optical depth of possible different atmospheres.

Figure 6 is the radiative temperature, not the atmospheric temperature. Look at the layer by layer absorption and emission. If we break the atmosphere into 100 layers, the optical depth of each layer is 0.03 making the emissivity equal 0.029554.

..τ… I+…. I-
3.00 600.0 360.0
2.97 596.4 356.4

The amount of radiation absorbed is then:

((600+596.4)/2+(360+356.4)/2)(1-exp(-0.03) = 28.266

The layer temperature is then:

((28.266/2)/0.029554)/5.67E-08)^0.25 = 303.05 K

The surface temperature is 320.7 K

This is the temperature discontinuity that will always be seen when inserting an arbitrary surface into the simple semi-gray atmosphere model. The surface has to have an effective temperature high enough to equal the effective temperature all the layers below if the surface weren’t present.

I have followed the authors output over the last few years and find bits of what he says plausible and other parts totally incomprehensible. There also seem to be some parts missing. His first language is not English and did he say in another forum that his knowledge of computers/spreadsheets was limited?

All in all it is a shame that he doesn’t find a suitable co author whose first language is Engish, is able to mathematically interpret what he says, argue with him when necessary and attempt to produce a coherent theory of a standard that could be presented for peer review. Wheher it would actually pass peer review is quite another matter of course.

Hey Science of Doom, look like you’ve just been volunteered…:)
(incidentally another great post)

On the specific issue of B0 and differentiating Eq 7, I understood that while B0 is constant over the range of τ for a given atmosphere, FM is differentiating with respect to τ_A – the total optical depth of possible different atmospheres.

I understood the same point. After all he is differentiating the equation with respect to τA to find out what relationships exist.

However, when you derive the SGM equation the integration constant is always F0, and has no dependence on τA.

So dB0/dτA = 0.

The value that does change in the SGM model with τA is BG – the source function for the ground. As τA increases the surface temperature increases.

SoD,
The thing is, FM has rejected this condition (in 15). He sets out his argument at the start of Sec 4.2. He notes that Eq 15 leads to a discontinuity at the bottom. Instead of the Eq 15 condition at the top, he uses his energy balance criterion, and takes BG as data (not depending on τ_A). Then there is a non-trivial dependence of B0 on τ_A.

I think he’s wrong. You can’t have a discrepancy at TOA, because there all heat transfer is radiative. You can in the lower troposphere, because there is convection (and clouds). So Eq 15 is the only solution that makes sense, and it applies from TOA down until the underlying equation (constant H) fails.

But his assumption is the basis for the algebra he does in Appendix B.

What I am saying is the fact that Miskolczi believes B0 is a variable does not make it so. He actually needs to prove it.

Especially when he has used the formula which proves it is a constant.

Miskolczi says (section 4.1, p.12&13):

Note, that in obtaining B0 , the fact of the
semi-infinite integration domain over the optical depth in the formal solution is widely used. For finite or optically thin atmosphere Eq. (15) is not valid. In other words, this equation does not contain the necessary boundary condition parameters for the finite atmosphere problem..

..Eq. (15) assumes that at the lower boundary the total flux optical depth is infinite. Therefore, in cases, where a significant amount of surface transmitted radiative flux is present in the OLR , Eqs. (16) and (17) are inherently incorrect.

[Emphasis added]

This is not true as already demonstrated.

From section 4.2, p13:

..The complete solution of Eq. (12) requires only one boundary condition. To evaluate B0 we can use either the top of the atmosphere or the surface boundary conditions since both of them are defined.

This is not true. The surface boundary condition is not defined. Not in this theory (the idealized semi-gray non-convective model).

And if “extension” of the theory results in a known boundary condition being violated – F↑ (OLR) ≠ F0 (absorbed solar), it demonstrates the “extended” theory wrong (as you point out in your comment).

And he doesn’t actually re-derive the equation.

Instead he substitutes values from the existing derivation into the general equation of radiative transfer and just claims that (somewhere along the line) B0 has now become a variable.

The semi-gray model is just a very simple model, not particularly useful for our atmosphere. Even if it could be re-derived to fit Miskolczi’s claims this still wouldn’t get anyone anywhere. And it still can’t be substituted into the general equation of radiative transfer.

And BG is still a variable, not a constant.

Nick, I’m sure we both agree – Miskolczi has assumed / claimed / imagined that B0 is variable. But that doesn’t make it so.

Here is the derivation of the SGM, referencing figure 2 in the article and the general Schwarzschild equation: dI/dτ=B-I.

-dF↑/dτ + F↑ = πB ….[1a]

dF↓/dτ + F↓ = πB ….[1b]

(The reason for the sign difference is direction of F↑ and F↓).

And the longwave fluxes, F↑, F↓ and B are functions of τ (i.e. they change with height in the atmosphere), but this is not written explicitly to make the equations easier to follow.

Net flux is a constant, by definition of radiative equilibrium there is zero heating of every layer.

⇒ F↑ – F↓ = constant

(For newcomers, why is it not zero? Because flux can pass through the atmosphere without interacting with it. But net reduction in upward flux has to be balanced by an increase in downward flux, otherwise energy is added to a layer).

As I remember, even moderate resolution band models are too computationally intensive for a GCM. RT is parameterized too. One of the CMIP studies found large discrepancies between more accurate RT models and those used in some of the GCM’s in the fairly recent past. There are still significant discrepancies between the AOGCM radiation codes and LBL codes in the models used for IPCC AR4.

WRT Eq 4. At last, an ‘offset C’ that can deal with ‘latency’. IMHO you’ll need at least two of these offsets, one to deal with ‘water vapour to water’ (and inverse) phase changes and one to deal with ‘water to ice’ (and inverse) phase changes (remembering that these phase changes occur closest to a ‘Cp’ definition). However, ice also sublimates.

You can’t simply throw convection, either sensible or latent, into a simple gray atmosphere model. I’ve tried and haven’t come close to making it work. I’m pretty sure you need a full bore radiative/convective model with a non-gray atmosphere, at least a band radiative model. I have some sort of 1D R-C model that came with a book on atmosphere modeling. I think it’s just Fortran source code, though. I have the nasty suspicion that convection is so highly parameterized that it amounts to a kludge. However, according to the description in the book, you can start with a warm or cold isothermal atmosphere and have it converge to a temperature profile that looks more or less like the Earth’s atmosphere.

This semi-grey model is very simple. It doesn’t include convection. Radiative equilibrium implies that temperatures have reached equilibrium via exchange of radiation only.

This isn’t the model that you want to build on. (Or develop a “new greenhouse theory” on).

GCM’s are much more comprehensive – although this article isn’t covering that subject. However, GCMs use the general solution described in the article and with band models because the line by line (LBL) solution requires too much computation. These band models are verified against LBL solutions of course.

Then GCMs also use various parameterizations to cover convection. A big subject and one for another day. There is a little on the performance of these models vs reality in Models On – and Off – the Catwalk

“This semi-grey model is very simple. It doesn’t include convection. Radiative equilibrium implies that temperatures have reached equilibrium via exchange of radiation only.”

Exactly the point. However, to look at this in another way. There needs to be a temperature difference within the depth to extinction for a net radiative transfer to occur, no? What if a change of phase is underway within that radiative shell?

Temperatures are buffered while the radiative flux soaks ‘into/out of’ latent heat because, loosely speaking, the latency due to phase change alters the energy capacity for the region. Isn’t this so?

Primarily, I don’t understand how a static model like the SGM can be used to evaluate M’s theory without modification to it. Once LTE is achieved the model is ‘dead’! IOW, any expansions/contractions, changes of phase for H20 and pressure changes have already occurred and the system is ‘torpid’ (unresponsive). However, Earth’s systems never achieve LTE (much to the disbelief of Tom Vonk) and are always changing diurnally, thus, any model that employs LTE as a requirement must be unrepresentative for the purposes of an Earth system.

All I’m trying to do here is improve the SGM’s ability to provide an improved representation of Earth’s atmosphere. I can only see this as an arrangement of arrays that record the offset of individual anomalies that present between the natural state and the state of LTE.

When you start talking about heats of vaporization and fusion, the implication is that you’re talking about convection, specifically latent heat transfer. The SGM doesn’t have clouds and even if you put in an absorber with a different scale height like water vapor, it’s never saturated, so there are no phase changes above the surface.

LTE (local thermal equilibrium) is different from TE (thermal equilibrium). LTE always applies for altitudes below 60 km. It means that collisional energy transfer between molecules dominates with the kinetic energy distribution of the gas molecules following the Boltzmann distribution. TE never happens in the real world or the SGM. The atmosphere would be isothermal if it did and heat wouldn’t be flowing in and out.

A consequence of LTE is that temperature measured by any means is the same. In plasma emission spectroscopy, LTE doesn’t exist, radiative energy loss is quite large, for example, and different methods of measuring temperature give different results.

Primarily, I don’t understand how a static model like the SGM can be used to evaluate M’s theory without modification to it.

Miskolczi’s theory is built on it. So evaluating Miskolczi’s theory requires evaluating the SGM.

Once LTE is achieved the model is ‘dead’! IOW, any expansions/contractions, changes of phase for H20 and pressure changes have already occurred and the system is ‘torpid’ (unresponsive).

However, Earth’s systems never achieve LTE (much to the disbelief of Tom Vonk) and are always changing diurnally, thus, any model that employs LTE as a requirement must be unrepresentative for the purposes of an Earth system.

DeWitt Payne has already commented on this. Many people confuse LTE with thermodynamic equilibrium because they sound the same.

LTE has a specific and technical meaning and it does not mean that the body in question is in thermal equilibrium.

Thanx guys and sorry, I confused TE with LTE. I should’ve left the post mentioning ‘LTE’ until the next day, but my head is full of stuff that I want to get out and don’t usually get the time to do so.

thus, I don’t believe that such a person could be so ‘naive’ as to present an SGM model that isn’t modified in some way that enables the model to simulate a ‘real sky’ and not just a ‘clear sky’. This is why I mentioned (in another thread) that ‘you’ll probably need Ferenc’s notes SoD’.

My presence here is not meant to be ‘a pain in the bum’, I want it to be a help. Thus, I ask you to consider the ‘offset C’ in Eq. 4 of SoD’s post of May 15, 2011 at 8:57 pm as an ‘array’ of energy that changes within the boundaries of a parcel, but doesn’t alter the total energy of that parcel. Thus, TE can be maintained for the parcel whilst energy transitions continue to be evolved within it. This would permit a ‘wet atmosphere’ where only a ‘dry atmosphere’ could previously exist.

Plug this equation into equation B1 (see above) and integrate from τ=0 to τ_A we should get πI+(0)= H. If I use πBg = H/2(1+τ_A+exp(-τ_A) (B11), H=240 and τ_A =1.8, the result is 188.483, a lot less than 240. Therefore as has been shown elsewhere, the solution is incorrect. You simply cannot force the temperature of the atmosphere at the surface to equal the surface temperature for the simple gray atmosphere model with a finite τ_A without violating another boundary condition.

By adding a window and a convective heat transfer term to the gray atmosphere model, I can produce something that looks somewhat like the real atmosphere, at least in the lower part. There’s no significant discontinuity in the temperature vs altitude plot at the surface and the initial lapse rate is ~5.7 K/km. I couldn’t do it with either alone. The window lets me increase τ_A enough to bring the atmosphere up in temperature and the convection term reduces the surface temperature and provides the excess energy for atmospheric emission that isn’t supplied by radiative absorption. This is for an atmosphere that is perfectly transparent to incoming short wave radiation.

But the point is still that d(Bo)/dτ is always zero, i.e. Bo is a constant. Without a theory that requires a constant optical depth or whatever ratio of fluxes to remain constant, all the messing about with NCEP/NCAR reanalysis numbers is completely meaningless. Those ratios only apply to conditions near present conditions, not to conditions where, say, CO2 is doubled.

SoD, I think you are right that there is a problem with combining the source function of the semi-gray atmosphere with the general radiative transfer equation.

But I think you misunderstood the purpose of taking the derivative of equation B7. Referring to “The most efficient cooling of the clear atmosphere requires a total optical depth that maximizes Bo”, you said “If someone wants to explain what thermodynamic principle create the first statement – I would be delighted.”

The global average relative humidity at 400 mbars is about 35%. What thermodynamic principle limits it to only 35%? This is much lower than the saturation limit, so that is not the reason. If it is not limited by a property of water (the saturation limit), then it must be limited by an energy restraint.

Miskolczi postulates that the amount of water vapour is determined by the requirement of the “maximum entropy principle” such that the optical depth adjusts to maximize the OLR, and Bo for any given surface temperature. He also postulates that the cloud cover is determined by the maximum entropy principle (page 19, M2007).

The Maximum entropy principle is an extension of the second law. It can be stated as “A system will select the path or assemblage of paths out of available paths that minimizes the potential or maximizes the entropy at the fastest rate given the constraints”. See:http://www.lawofmaximumentropyproduction.com/

This implies that the atmosphere will maximize the Bo by adjusting water vapour amount for any given Bg.

Instead of saying the amount of GHG is what is currently there plus what we add plus feedbacks, we say the amount of GHG is that which maximizes the Bo for each possible surface temperature. This requires calculating dBo/dtau, keeping Bg constant. Therefore, in equation B10, your “correction” in red is not required.

One problem is that B0 is a constant which is determined – and required – by the First Law of Thermodynamics.

If a solution conflicts with this important law I think we can all agree that the solution is in error?

So the fundamental step (assuming that there was a point to developing the source function for the semi-gray atmosphere within the general RTE solution..) would be to demonstrate why B0 varying didn’t violate the first law of thermodynamics. That will be difficult, I think.

The global average relative humidity at 400 mbars is about 35%. What thermodynamic principle limits it to only 35%? This is much lower than the saturation limit, so that is not the reason. If it is not limited by a property of water (the saturation limit), then it must be limited by an energy restraint.

The conclusion doesn’t follow at all. But the topic is not a simple one.

Convection takes parcels of air upwards – and if this was the only process then the relative humidity (above the boundary layer) would be at 100%.

But air rising is balanced by air that is descending. In fact, in the tropics deep convection accounts for ≅ 10% of the area, and subsiding air accounts for ≅ 90%. (The ascending air is moving a lot faster than the descending air so mass balance does occur).

The descending air was saturated at its point of coldest ascent. So as it descends its relative humidity drops – and in the sub-tropics this can result in relative humidity less than 10%.

Air mixes. So ascending air and descending air produce an average of relative humidity decreasing with height. The observed features can be approximately produced by climate models. You can see an example in the Held & Soden paper.

Bo must be a constant because the math requires it. When you integrate a function, say x, the result is the integral of the function plus a constant.

∫xdx = x²/2 + C

C must always be constant because otherwise the derivative of the integral wouldn’t be equal to the original function.

d(x²/2 + C)dx = x

dC/dx ≡ 0

Yet Miskolczi in Appendix B of M2007 treats the integration constant Bo as if it were a variable and treats the variable Bg as if it were a constant. You can’t do that and claim your results are correct. It’s the equivalent of saying 2 + 2 = whatever I want it to be at any given time.

If you actually do the calculations based on Miskolczi’s equations, the results violate the original boundary conditions.

I decided to move to this post as it seemed more appropriate. Usign the typical radiative interchange diagrams like figure 2 above, and extending to multiple layers, I came up with equations for different formulas for tranmissibility. The first is one in which the transmissibility is e^(-ko*dl). I come up with formulas for each layer, and then equations as the limit when the number of layers goes to infinity. I then expanded the analysis for a transmissibility like exp(-ko*rho*dl), where rho changes with P and T. It turned out that by choosing a coordinate in mass all the equations are the same, except an auxiliary equation is needed to get the height. These solutions are close form, exact and contain no approximations, but based on several assumption: transmitability as described, no reflectivity, ideal gas law, constant gravity, semigrey model (clear in shortwave, grey in longwave, a constant emissivity across the spectrum), plane-parallel model, no convection and only looking straight up.

It turns out that the Radiative terms all can be ratioed by the incoming long wave radiation So*(1-A)/4 which in equilibrium is equal to OLR. I did all my analysis with a ‘total’ emissivity et, but in terms of total optical thikness tau it is:
tau = -ln(1-et) or et = 1-e^-tau

I went ahead and tried to cast my results in the variables provided by Milkolczi, here are some:

We see that there is a discontinuity from the ground temperature to the bottom air temperature (no convection), that there is a finite top atmosphere temperature, that the mentioned temperatures and the greehouse factor are completely defined by the optical thickness (so tau is equivalent to GH). Also we see that AA is not equal to ED.

Now, by having an auxiliary equation for height, it turns out that if the boundary condition for the the last layer is zero pressure, then that layer becomes of infinite height (I would put an arbitrarily small pressure as boundary to avoid that) so I guess this would be an infinite atmosphere, granted with miniscule zero density and pressure at the top. The part I still need to figure out is were the term exp(-tau) in the equation f = 1/2 + tau/2 + exp(-tau) comes from and how does it relate to my model, as it seems to be key in Miskolczi’s derivation, bringing AA ~ ED as well as bringing the bottom atmospheric temperature equal to ground temp.

Any information on the difference between his ‘semi-infinite atmosphere’ and his ‘bounded or semi-tranparent’ atmosphere, and physical meening of the extra term will be appreciated it.

Ok I know there is a complicated derivation for it, but I did not get my results from any of those formulas, just by the application of transmissibility = exp(-ko* density * delta length) in exact closed form. I would like to see if I can reproduce his equation using my method.

The part I still need to figure out is were the term exp(-tau) in the equation f = 1/2 + tau/2 + exp(-tau) comes from…

It comes from Miskolczi’s incorrect derivation in Appendix B of M2007. If you (properly) treat Bo as a constant and Bg as a variable, you get a different value. It’s all there in the comment thread somewhere.

Yes, but how can I argue with results? Figures 21 and 22 of “The greenhouse effect and the spectral decomposition of the clear-sky terrestrial radiation” by Miskolczi and Mlynczak compares the various ways to calculate the ground temp and skin temp, and at first sight, adding the extra exp(-tau) really seems to improve things. That would be no different than adding a term or constant heuristically, and seeing if it improves a correlation, regardless of how it was come about. I just have not been able to reproduce it.

In any event, I thought that if I made the ground temperature equal to the bottom atmospheric layer temperature to remove the discrepancy, then I could get closer to that equation. So I added a K term taking energy from the ground and having it completely absorbed in the bottom layer. And I was still able to get an exact result. I still find it hard to believe, but I guess that how the math works out.

To my surprise, the ground temperature did not change. It was completely invariant to convection. The bottom air layer temp did change and became equal to the ground temp, and the temp profile changed, but the top layer temp also remained constant. Amazingly this means that you could have an adiabatic rate and the ground temperature would still be correctly estimated from a model in radiative equilibrium with no convection. I used to think that maybe one could find the height of the tropopause first and then work downwards to where the ground temperature would be from the lapse rate, but it looks like its the other way around: The ground temp is defined by the total optical thickness only, and then the temp decreases from that fixed point upwards towards the tropopause.

SOD: I’ve meaning to ask you, in your rte program you adjusted the height to equipartition the pressure.. why did you make that choice? did it help in some calculation?. I just realized that dP = -rho*g*dz is dP=-dm/Area*g or basically you equipartion the mass just as I did.. just curious.

Argh.. No wonder it did not make sense, I was wrong. Convection from the ground will change the ground temp. What happened was that my convection went only into the first layer to match the temps, then as the first layer got smaller and smaller the convection got smaller and smaller until it was zero and had no effect. I think I know how to fix it.

Trying it the other way, by setting K so that AA=ED brought the temperatures at the ground into near equality. So I think I’m onto something here.
In any event, posting like this and having to correct my self at every turn is turning into a novel. I will try to prepare a draft of a paper of the semi-grey model I am developing and post it into a site, that way I can make revisions to it as it gets tested and validated. It just takes time and a lot of work. Hopefully something useful will come of this.